Learnable Multipliers: Freeing the Scale of Language Model Matrix Layers

TL;DR

This paper introduces learnable scalar multipliers for matrix layers in language models to optimize weight scale, outperforming fixed scaling methods and improving downstream task performance.

Contribution

It proposes a novel learnable scaling approach for matrix layers, generalizing muP multipliers, and demonstrates performance gains over fixed and baseline methods.

Findings

Learnable scalers adapt to data and improve model performance.

Outperforms well-tuned muP baseline in experiments.

Shows consistent improvements with different optimizers.

Abstract

Applying weight decay (WD) to matrix layers is standard practice in large-language-model pretraining. Prior work suggests that stochastic gradient noise induces a Brownian-like expansion of the weight matrices W, whose growth is counteracted by WD, leading to a WD-noise equilibrium with a certain weight norm ||W||. In this work, we view the equilibrium norm as a harmful artifact of the training procedure, and address it by introducing learnable multipliers to learn the optimal scale. First, we attach a learnable scalar multiplier to W and confirm that the WD-noise equilibrium norm is suboptimal: the learned scale adapts to data and improves performance. We then argue that individual row and column norms are similarly constrained, and free their scale by introducing learnable per-row and per-column multipliers. Our method can be viewed as a learnable, more expressive generalization of muP multipliers. It outperforms a well-tuned muP baseline, reduces the computational overhead of multiplier tuning, and surfaces practical questions such as forward-pass symmetries and the width-scaling of the learned multipliers. Finally, we validate learnable multipliers with both Adam and Muon optimizers, where it shows improvement in downstream evaluations matching the improvement of the switching from Adam to Muon.